Einstein solvmanifolds: existence and non-existence questions

Jorge Lauret; Cynthia Will

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arXiv · math/0602502

Einstein solvmanifolds: existence and non-existence questions

Jorge Lauret·Cynthia Will

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Abstract

The general aim of this paper is to study which are the solvable Lie groups admitting an Einstein left invariant metric. The space N of all nilpotent Lie brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds, containing the set of all those which are Einstein in that dimension. The moment map for the natural GL(n)-action on N evaluated in a point of N encodes geometric information on the corresponding solvmanifold, allowing us to use strong and well-known results from geometric invariant theory. For instance, the functional on N whose critical points are precisely the Einstein solvmanifolds is the square norm of this moment map. We also use a GL(n)-invariant stratification for the space N following essentially a construction given by F. Kirwan and show that there is a strong interplay between the strata and the Einstein condition on the solvmanifolds. As applications, we obtain several examples of graded (even 2-step) nilpotent Lie algebras which are not the nilradicals of any standard Einstein solvmanifold, as well as a classification in the 7-dimensional 6-step case and an existence result for certain 2-step algebras associated to graphs.

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BibTeX RIS

Jorge Lauret, Cynthia Will. 2006-02-22. Einstein solvmanifolds: existence and non-existence questions. https://arxiv.org/abs/math/0602502

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